Not sure why people are referring to travel on six different days, the JR Tohoku-South Hokkaido Rail Pass is only valid for five travel days. That is simply another pass.
The pass here asked about is JR East-South Hokkaido Rail Pass,
which is usable also in and around Tokyo,
sold at 26000 yen / 27000 yen outside / within Japan,
and valid for the 6 selected days during the 14 days from the day of issuance.
The six days for this pass are not necessarily consecutive.
http://www.jreasthokkaido.com/easthokkaido/e/Your itinerary's Sapporo -- Asahikawa leg
is mostly outside the validity area of this pass,
with only the section Sapporo -- Shiroishi covered.
A combination of this pass and a discount ticket (for your Asahikawa trip)
sounds suitable for your itinerary.
that round trip is almost 10,000 yen by Limited Express (reserved seating) that covers the difference in price between the two passes. The difference in the price is not the difference in the actual cost.
To obtain a (7-day Ordinary Class) Japan Rail Pass, you pay
a handling fee
+ a transportation fee to visit the sales office or a delivery fee
in addition to the pass itself (about 29000 yen).
The 7 days for your Japan Rail Pass are necessarily consecutive.
When you purchase a JR East-South Hokkaido Rail Pass,
as long as you can visit JR East Travel Service Center in HND
on Day 1 during the business hours (6:45 - 20:00),
you pay just for the pass itself (27000 yen).
Round-trip Non-Reserved Seating Discount Ticket (S Ticket)
: 5080 yen for Limited Express train rides from Sapporo to Asahikawa and back without seat reservation.
http://www2.jrhokkaido.co.jp/global/pdf/e-direction03.pdfAsahiyama Dobutsuen Kippu
: 6130 yen / 7170 yen for
Limited Express train rides from Sapporo to Asahikawa and back
without seat reservation / with your seats reserved
+ bus rides from Asahikawa Sta. to Asahiyama Zoo and back
+ admission to Asahiyama Zoo.
http://www2.jrhokkaido.co.jp/global/pdf/e-asahiyama201904.pdf